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There is no problem in Landau's book. The question appears to be based on a misunderstanding.

The Gibbs distribution provides the probability of finding a quasi-closed subsystem (whose size is much smaller than the whole system and closed in the sense that there are no interactions with other parts of the full system; if the subsystem is macroscopic then the interactional effects with the other parts of the system can be neglected during the quasi-closed stage) to be in a particular quantum state with energy $E_n$ where $n$ denotes the set of all quantum numbers pertaining to the quantum state of interest, meaning that $E_n$ is a microscopic description of the state, while at the same time ignoring the microscopic description of the remaining part of the system (referred to as the medium or heat bath), that is to say, the medium is in some macroscopically defined state. In summary, Gibbs distribution can be applied to any subsystem as long as it is quasi-closed (i.e., no interactions with its neighbors) and smaller than the full system. (Gibbs distribution can also be applied to the full system in an approximate sense. This interesting point is discussed in the last two paragraphs of section 28 in Landau and Lifshitz's book)

In a perfect gas, by definition, interactions are neglected, so that although individual molecules are not macroscopic, they can be considered quasi-closed. Thus, the Gibbs distribution is applicable to individual molecules in an ideal gas.

There is no problem in Landau's book. The question appears to be based on a misunderstanding.

The Gibbs distribution provides the probability of finding a quasi-closed subsystem (whose size is much smaller than the whole system and closed in the sense that there are no interactions with other parts of the full system; if the subsystem is macroscopic then the interactional effects with the other parts of the system can be neglected during the quasi-closed stage) to be in a particular quantum state with energy $E_n$ where $n$ denotes the set of all quantum numbers pertaining to the quantum state of interest, meaning that $E_n$ is a microscopic description of the state, while at the same time ignoring the microscopic description of the remaining part of the system (referred to as the medium or heat bath), that is to say, the medium is in some macroscopically defined state. In summary, Gibbs distribution can be applied to any subsystem as long as it is quasi-closed (i.e., no interactions with its neighbors) and smaller than the full system. (Gibbs distribution can also be applied to the full system without any difficulty because of the reasons stated in the last two paragraphs of section 28 in Landau and Lifshitz's book)

In a perfect gas, by definition, interactions are neglected, so that although individual molecules are not macroscopic, they can be considered quasi-closed. Thus, the Gibbs distribution is applicable to individual molecules in an ideal gas.

There is no problem in Landau's book. The question appears to be based on a misunderstanding.

The Gibbs distribution provides the probability of finding a quasi-closed subsystem (whose size is much smaller than the whole system and closed in the sense that there are no interactions with other parts of the full system; if the subsystem is also macroscopic then the interactional effects with the other parts of the system can be neglected during the quasi-closed stage) to be in a particular quantum state with energy $E_n$ where $n$ denotes the set of all quantum numbers pertaining to the quantum state of interest, meaning that $E_n$ is a microscopic description of the state, while at the same time ignoring the microscopic description of the remaining part of the system (referred to as the medium or heat bath), that is to say, the medium is in some macroscopically defined state. In summary, Gibbs distribution can be applied to any subsystem as long as it is quasi-closed (i.e., no interactions with its neighbors) and smaller than the full system. (Gibbs distribution can also be applied to the full system in an approximate sense. This interesting point is discussed in the last two paragraphs of section 28 in Landau and Lifshitz's book)

In a perfect gas, by definition, interactions are neglected, so that although individual molecules are not macroscopic, they can be considered quasi-closed. Thus, the Gibbs distribution is applicable to individual molecules in an ideal gas.

There is no problem in Landau's book. The question appears to be based on a misunderstanding.

The Gibbs distribution provides the probability of finding a quasi-closed subsystem (whose size is much smaller than the whole system and closed in the sense that there are no interactions with other parts of the full system; if the subsystem is macroscopic then the interactional effects with the other parts of the system can be neglected during the quasi-closed stage) to be in a particular quantum state with energy $E_n$ where $n$ denotes the set of all quantum numbers pertaining to the quantum state of interest, meaning that $E_n$ is a microscopic description of the state, while at the same time ignoring the microscopic description of the remaining part of the system (referred to as the medium or heat bath), that is to say, the medium is in some macroscopically defined state. In summary, Gibbs distribution can be applied to any subsystem as long as it is quasi-closed (i.e., no interactions with its neighbors) and smaller than the full system. (Gibbs distribution can also be applied to the full system in an approximate sense. This interesting point is discussed in the last two paragraphs of section 28 in Landau and Lifshitz's book)

In a perfect gas, by definition, interactions are neglected, so that although individual molecules are not macroscopic, they can be considered quasi-closed. Thus, the Gibbs distribution is applicable to individual molecules in an ideal gas.

There is no problem in Landau's book. The question appears to be based on a misunderstanding.

The Gibbs distribution provides the probability of finding a quasi-closed subsystem (whose size is much smaller than the whole system and closed in the sense that there are no interactions occur with other parts of the full system; if the subsystem is also macroscopic then the interactional effects with the other parts of the system can be neglected during the quasi-closed stage) to be in a particular quantum state with energy $E_n$ where $n$ denotes the set of all quantum numbers pertaining to the quantum state of interest, meaning that $E_n$ is a microscopic description of the state, while we are not interested in the microscopic description of the remaining part of the system (referred to as the medium or heat bath), that is to say, the medium is in some macroscopically defined state. In summary, Gibbs distribution can be applied to any subsystem as long as it is quasi-closed (i.e., no interactions with its neighbors) and smaller than the full system. (Gibbs distribution can also be applied to the full system. This interesting point is discussed in the last two paragraphs of section 28 in Landau and Lifshitz's book)

In a perfect gas, by definition, interactions are neglected, so that although individual molecules are not macroscopic, they can be considered quasi-closed. Thus, the Gibbs distribution is applicable to individual molecules in an ideal gas.

There is no problem in Landau's book. The question appears to be based on a misunderstanding.

The Gibbs distribution provides the probability of finding a quasi-closed subsystem (whose size is much smaller than the whole system and closed in the sense that there are no interactions with other parts of the full system; if the subsystem is also macroscopic then the interactional effects with the other parts of the system can be neglected during the quasi-closed stage) to be in a particular quantum state with energy $E_n$ where $n$ denotes the set of all quantum numbers pertaining to the quantum state of interest, meaning that $E_n$ is a microscopic description of the state, while at the same time ignoring the microscopic description of the remaining part of the system (referred to as the medium or heat bath), that is to say, the medium is in some macroscopically defined state. In summary, Gibbs distribution can be applied to any subsystem as long as it is quasi-closed (i.e., no interactions with its neighbors) and smaller than the full system. (Gibbs distribution can also be applied to the full system in an approximate sense. This interesting point is discussed in the last two paragraphs of section 28 in Landau and Lifshitz's book)

In a perfect gas, by definition, interactions are neglected, so that although individual molecules are not macroscopic, they can be considered quasi-closed. Thus, the Gibbs distribution is applicable to individual molecules in an ideal gas.